In order to discuss the Raman active modes for two-phonon Raman scattering, we first make analysis on group
theoretical selection rules for two-phonon scattering. Due to the dominant \(ee\)  and \(hh\)  processes for the DRR,
the \(\mu _{th}\)  and \(\nu _{th}\)  two phonons contributing to a single DRR process should have the same wave vector
with opposite sign to each other, q\(_{\mu }\)  and -q\(_{\nu }\) . The origin of DRR which can arise from either
combination, or subtraction, or overtone of two phonon modes can therefore be analyzed at the same \(q\)  point
in the phonon dispersion relation in Fig. REF (b). The point groups along high-symmetry line and \(q\) 
points are put on top, such as point group D\(_{3h}\)  for the highest symmetric
zone-center \(\Gamma \)  point, C\(_{3h}\)  for the zone corner \(K\)  point, C\(_{2v}\)  for the zone-edge middle \(M\)  point
and between the \(\Gamma \)  and \(M\)  points, and C\(_s\)  for the rest of \(q\)  points (including between the \(\Gamma \) 
(or \(M\) ) and \(K\)  points) in the BZ. The irreducible representations are also given to the corresponding phonon
bands. Three typical zone-center Raman-active modes (A\(^{^{\prime }}_1\) , E\(^{^{\prime }}\)  and E\(^{^{\prime \prime }}\) ) are visualized on the
right of Fig. REF (b) with the Raman tensors. By the general methods of Birman [1]}, [2]},
with the help of the Bilbao crystallographic server [3]}, we can determine whether two-phonon modes
for a \(q\)  point are Raman-active or not, by correlating the irreducible representations of the combined modes of
the group at the \(q\)  point with the irreducible representations of the full space group (D\(_{3h}\) ) at the \(\Gamma \) 
point. To ascertain the Raman activity of a two-phonon DRR mode, the reduced irreducible constituents from the
Kronecker products have to contain at least one of the three Raman-active symmetries (A\(^{^{\prime }}_1\) , E\(^{^{\prime }}\)  and E\(^{^{\prime \prime }}\) )
of zone-center point group D\(_{3h}\) . In the supplementary Table 1, we list all the possible combinations
of two phonons in the BZ and the reduction of symmetries, and mark those Raman-active ones in blue color. First
of all, the overtone modes, even from two Raman-inactive symmetries, are all Raman active in the whole BZ, since
the decomposed symmetries include at least Raman-active A\(^{^{\prime }}_1\)  symmetry. Second, for \(q\)  point beyond the
zone-center \(\Gamma \)  point (at \(M\)  or between \(\Gamma \)  and \(M\)  of C\(_{2v}\)  group), especially with lower symmetry
(C\(_s\)  group), the decomposed symmetries usually have two Raman-active symmetries (A\(^{^{\prime }}_1\)  + E\(^{^{\prime }}\) ) coexisting.
Considering that A\(^{^{\prime }}_1\)  and E\(^{^{\prime }}\)  have different circular polarization selectivity [4]}, we may
anticipate that DRR modes at these points should have non-zero Raman intensity in both \(\sigma _{+}\sigma _{+}\)  and
\(\sigma _{+}\sigma _{-}\)  configurations, which can be used to test some possible two-phonon assignments.
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